Glossy material affecting color of another objects
Edit: 

My bad, your cube is actually opaque, which was hard to guess from the pictures, I don't know if it's feasible with the Light Path node (?) so, as Thedigitalzorn said, the solution is probably to use the Compositor.

Another simple trick would be to launch a render of the arrow only, in the Properties panel > Render > Film, enable Transparent to make sure that the video has a transparent background. Then render the cube and the setup. At last, stack the 2 sequences in the Video Sequencer. 

Old answer:

Give your arrow an Emission node instead of a Diffuse. And if you don't want it to emit any light, deactivate its Diffuse option in the Properties panel > Object > Visibility.